approximate coefficients, which is a highly unstable problem. The first issue has been solved by Dahan and Schost: Among the sets of regular chains that represent a given set of solutions, there is a set for which the coefficients are explicitly bounded in terms of the size of the input system, with a nearly optimal bound. This set, called equiprojectable decomposition, depends only on the choice of the coordinates. This allows the use of modular methods for computing efficiently the equiprojectable decomposition. The second issue is generally solved by outputting regular chains of a special form, sometimes called shape lemma, for which all but the first one are equal to . For getting such regular chains, one may have to add a further variable, called separating variable, which is given the index . The rational univariate representation, described below, allows computing such a special regular chain, satisfying Dahan–Schost bound, by starting from either a regular chain or a Gröbner basis. Rational univariate representation The rational univariate representation or RUR is a representation of the solutions of a zero-dimensional polynomial system over the rational numbers which has been introduced by F. Rouillier. A RUR of a zero-dimensional system consists in a linear combination of the variables, called separating variable, and a system of equations where is a univariate polynomial in of degree and are univariate polynomials in of degree less than . Given a zero-dimensional polynomial system over the rational numbers, the RUR has the following properties. All but a finite number linear combinations of the variables are separating variables. When the separating variable is chosen, the RUR exists and is unique. In particular and the are defined independently of any algorithm to compute them. The solutions of the system are in one-to-one correspondence with the roots of and the multiplicity of each root of equals the multiplicity of the corresponding solution. The solutions of the system are obtained by substituting the roots of in the other equations. If does not have any multiple root then is the derivative of . For example, for the system in the previous section, every linear combination of the variable, except the multiples of , and , is a separating variable. If one chooses as a separating variable, then the RUR is The RUR is uniquely defined for a given separating variable, independently of any algorithm, and it preserves the multiplicities of the roots. This